# 5th Round of 18th year

### 1. ... a present form Buffalo Bill( points)

Buffalo Bill is trying to catch Jessie James (well known bandit). They met in town known as Clay County and the shooting started. Buffalo noticed a barrel full of petrol on the trolley between him and Jessie. How he should sent this barrel towards Jessie and set it on fire?

Jessie shot a hole in the 9/10 of the height and the petrol started to splash out of the barrel. Buffalo hit the barrel in the middle and shoots again. Calculate what acceleration will take the trolley depending on the place of the third hole in the barrel. Assume the momentum of the bullet is negligible and friction is negligible as well. Think about other interesting aspects of this match.

### 2. ... falling from the stairs(0 points)

Karel is playing with a ball. While rolling it on the floor is comes to he inclined plane, which serves as a staircase, and starts to slide down. The ball is moving in such direction, that the vector of its velocity $**v**$ and the top edge of the inclined plane shows and angle $φ$. Calculate a vector of the velocity $**v**′$ of the ball under the inclined plane (its magnitude and the direction), if the height of the plane is $h$. The friction is negligible, assume that the top edge is smooth so the ball will always follow the surface.

As a bonus: what is the difference of the direction of the ball falling into a cylindrical hole of radius $R$ and the depth $h$ with inclined sides (see figure 1). The length of the inclined wall can be neglected with respect to the overall size of the hole.

### 3. ... beta decay( points)

When measuring decay of neutron to electron and proton the energy of the electron was detected. How can be detected, that another particle was not created? Assume the neutron to be at rest at the beginning.

### 4. ... misbehaving gravity( points)

During the long term observations of Jupiter's moon Io it was noticed that the the orbit time of the Io around Jupiter (e.g. time between two consecutive 'disappearances' of the moon behind Jupiter) is oscillating regularly between 42 h 28 min 21 s and 42 h 28 min 51 s (with the measurement error of 2 s).

Try to explain qualitatively and quantitatively observed changes. By quantity we mean to determine the magnitude of the source of this oscillations from the measurements. And of course including the estimation of the experimental error.

### P. ... faster than water( points)

Is it possible for the boat to move faster than the water in the river? Justify you answer and assume laminar flow.

### E. ... flash-point( points)

Measure the flash-point (the lowest temperature at which the substance ignite) of the refill for the cigarette lighter. As a bonus measure the flash-point for the ethanol or other organic compound.

For example use a resistive wire warmed up by the electric current and let the flow of the gas from the lighter to flow over it. To calculate flash-point use values of the voltage and current and known temperature dependence of the resistivity. Do not restrict yourself to this suggested experiment!

Warning: Fire can cause serious burns, please proceed with extreme caution!

### S. ... Mercury, the pit and the pendulum(0 points)

The following questions will test the knowledge from all presented chapters about mechanics – Newtons formalism, D'Alembert's principle and Lagrange's formalism.

- Imagine planet Mercury orbiting around Sun. It is know, that its elliptic trajectory is rotating, the position of perihelion is moving, which cannot be explained by gravitation force.

$**F**~$=$~κ~$($mM**r**$) ⁄$~r$^{3}$~.$

Proof, that adding an additional central force

$**F**~$=$~C~$($**r**$) ⁄$~r$^{4}$~,$

where $C$ is suitable constant, full trajectory (ellipse) will rotate at constant angular speed. In other words, that exists a frame rotating at constant speed, where the trajectory is an ellipse. Knowing this angular speed $Ω$, calculate the constant $C$. Is such correction for gravitation enough?

- Calculate equilibrium position of homogeneous rod of length $l$ supported by inner wall of excavation in the V-shape (see figure 12) as a function of the angle of V-shape $α$.
- Using Lagrange's equations calculate period of small oscillations of double-reverse pendulum in image 13. The weight are at the ends of weightless rod of the length $l$ and have masses $m$_{1} and $m$_{2}, the distance from the joint from the weight $m$_{1} is $l$_{0}.